非线性系统及其绝妙的数学结构(第2卷)(英文)

本书包含了17篇由非线 性系统不同方面的顶尖专家 撰写的特邀论文，包括常微 分方程和偏微分方程、差分 方程、离散或晶格方程、非 交换方程和矩阵方程以及时 滞方程。本书内容被分为三 个主要部分：A部分为可积 性，Lax对与对称性，包含7 篇论文，作者主要解决了如 何检测可积系统的基本问题 ，以及非线性系统对称方法 的使用问题；B部分为代数 与几何方法，包含7篇论文 ，作者描述了获得非线性系 统显式解和（或）描述系统 解的结构的不同方法；C部 分为应用，包含了三篇论文 ，作者为了解决特殊的非线 性问题应用了不同的方法。

Preface The Authors A1. Reciprocal transformations and their role in the integrability and classification of PDEs 1. Introduction 2. Fundamentals 3. Reciprocal transformations as a way to identify and classify PDEs 4. Reciprocal transformations to derive Lax pairs 5. A Miura-reciprocal transformation 6. Conclusions A2. Contact Lax pairs and associated (3+l)-dimensional integrable dispersionless systems 1. Introduction 2. Isospectral versus nonisospectral Lax pairs 3. Lax representations for dispersionless systems in (I+I)D and (2+1)D 4. Lax reprcsentations for dispersionless systems in (3+l)D 5. R-matrix approach for dispersionless systems with nonisospectral Lax representations A3. Lax pairs for edge-constrained Boussinesq systems of partial difference equations 1. Introduction 2. Gauge equivalence of Lax pairs for PDEs and PAEs 3. Derivation of Lax pairs for Boussinesq systems 4. Gauge and gauge-like equivalences of Lax pairs 5. Application to generalized Hietarinta systems 6. Summary of results 7. Software implementation and conclusions A4. Lie point symmetries of delay ordinary differential equations 1. Introduction 2. Illustrating example 3. Formulation of the problem for first-order DODEs 4. Construction of invariant first-order DODSs 5. First-order linearDODSs 6. Lie symmetry classification of first-order nonlinear DODSs 7. Exact solutions of the DODSs 8. Higher order DODSs 9. Traffic flow micro-model equation 10. Conclusions A5. The symmetry approach to integrability: recent advances 1. Introduction 2. The symmetry approach to integrability 3. Integrable non-abelian equations 4. Non-evolutionary systems A6. Evolution of the concept of h-symmetry and main applications 1. Introduction 2. Basic notions on Lie point symmetries and C∞- symmetries of ODEs 3. Analytical applications of C~-symmetries 4. Extensions and geometric interpretations of C∞-symmetries A7. Heir-equations for partial differential equations: a 25-year review 1. Introduction 2. Constructing the heir-equations 3. Symmetry solutions of heir-equations 4. Zhdanov's conditional Lie-B~cklund symmetries and heir-equations